Shifts of Finite Type As Fundamental Objects in the Theory of Shadowing

SHIFTS OF FINITE TYPE AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING

CHRIS GOOD AND JONATHAN MEDDAUGH

Abstract. Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let X be a compact totally disconnected space and f : X → X a continuous map. We demonstrate that f has shadowing if and only if the system (f, X) is (conjugate to) the inverse limit of a directed system satisfying the Mittag- Leffler condition and consisting of shifts of finite type. In particular, this implies that, in the case that X is the Cantor set, f has shadowing if and only if (f, X) is the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type. Moreover, in the general compact metric case, where X is not necessarily totally disconnected, we prove that f has shadowing if (f, X) is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits.

1. Introduction Given a finite set of symbols, a shift of finite type consists of all infinite (or bi-infinite) symbol sequences, which do not contain any of a finite list of forbidden words, under the action of the shift map. Shifts of finite type have applications across mathematics, for example in Shannon’s theory of information [26] and sta- tistical mechanics. In particular, they have proved to be a powerful and ubiquitous tool in the study of hyperbolic dynamical systems. Adler and Weiss [1] and Sinai [31], for example, obtain Markov partitions for hyperbolic automorphisms of the torus and Anosov diffeomorphisms respectively, allowing analysis via shifts of finite type. Generalising the notion of Anosov diffeomorphisms, Smale [33] isolates subsystems conjugate to shifts of finite type in certain Axiom A diffeomorphisms. His fundamental example of a horseshoe, conjugate to the full shift space on two symbols, captures the chaotic behaviour of the diffeomorphism on the nonwandering set where the map exhibits hyperbolic behaviour. Bowen [6] then shows that the nonwandering set of any Axiom A diffeomorphism is a factor of a shift of finite type. In fact, shifts of finite type appear as horseshoes in many systems both hyperbolic (for example [34, 36]) and otherwise [18].

2000 Mathematics Subject Classification. 37B20, 54H20. Key words and phrases. discrete dynamical system, inverse limits, pseudo-orbit tracing, shadowing, shifts of finite type. The authors gratefully acknowledge support from the European Union through funding the H2020-MSCA-IF-2014 project ShadOmIC (SEP-210195797). The first author gratefully acknowl- edges the support of the Institut Mittag-Leffler, through the workshop Thermodynamic Formalism Applications to Geometry, Number Theory, and Stochastics. 1 2 C. GOOD AND J. MEDDAUGH

For a map f on a metric space X, a sequence hxiii∈ω is a δ-pseudo-orbit if d(f(xi), xi+1) < δ. Pseudo-orbits arise naturally in the numerical calculation of orbits. It turns out that pseudo-orbits can often be tracked within a specified toler- ance by real orbits, in which case f is said to have the shadowing, or pseudo-orbit tracing, property. Clearly this is of importance when trying to model a system numerically (for example [9, 10, 20, 21]), especially when the system is expanding and errors might grow exponentially (indeed shadowing follows from expansivity for open maps [25], see also [23]). However, shadowing is also of theoretical importance and the notion can be traced back to the analysis of Anosov and Axiom A diffeomorphisms. Sinai [32] isolated subsystems of Anosov diffeomorphisms with shadowing and Bowen [5] proved explicitly that for the larger class of Axiom A diffeomorphisms, the shadowing property holds on the nonwandering set. However, Bowen [6] had already used shadowing implicitly as a key step in his proof that the nonwandering set of an Axiom A diffeomorphism is a factor of a shift of finite type. The notion of structural stability of a dynamical system was instrumental in the definitions of both Anosov and Axiom A diffeomorphisms [33] and shadowing plays a key role in stability theory [22, 24, 35]. Shadowing is also key to characterizing omega-limit sets [2, 5, 19]. Moreover, fundamental to the current paper is Walters’ result [35] that a shift space has shadowing if and only if it is of finite type. In this paper we prove that there is a deep and fundamental relationship between shadowing and shifts of finite type. It is known that shadowing is generic for homeomorphisms of the Cantor set [3] and that the shifts of finite type form a dense subset of the space of homeomorphisms on the Cantor set [27]. Hirsch [17] shows that expanding differentiable maps on closed manifolds are factors of the full one sided shift. In [6], Bowen considers the induced dynamics on the shift spaces associated with Markov partitions to show that the action of an Axiom A diffeomorphism on its non-wandering set is a factor of a shift of finite type. Here we expand the scope of this type of analysis by considering the actions induced by f on shift spaces associated with several arbitrary finite open covers of the state space X, rather than the much more specific Markov partitions. In doing so, we are able to extend and clarify these results significantly, proving the following.

Theorem 18. Let X be a compact, totally disconnected Hausdorff space. The map f : X → X has shadowing if and only if (f, X) is conjugate to the inverse limit of an inverse system satisfying the Mittag-Leffler condition and consisting of shifts of finite type.

Corollary 19. Let X be the Cantor set, or indeed any compact, totally disconnected metric space. The map f : X → X has shadowing if and only if (f, X) is conjugate to the inverse limit of a sequence satisfying the Mittag-Leﬄer condition and consisting of shifts of ﬁnite type.

Let X and Y be compact metric spaces and φ : X → Y be a factor map between the systems f : X → X and g : Y → Y (so that φf(x) = gφ(x)). We say that φ almost lifts pseudo-orbits (φ is ALP) if and only if for all > 0 and η > 0, there exists δ > 0 such that for any δ-pseudo-orbit hyii in Y , there exists an η-pseudo- orbit hxii in X such that d(φ(xi), yi) < . This notion is also well defined in general Hausdorff spaces (Definition 24). SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 3

Theorem 26. Let X be a compact Hausdorff space. The map f : X → X has shadowing if (f, X) lifts via a map which is ALP to the inverse limit of an inverse system satisfying the Mittag-Leffler condition and consisting of shifts of finite type. Corollary 28. Let X be a compact metric space. The map f : X → X has shadowing only if (f, X) lifts via a map which is ALP to an inverse limit of a sequence of shifts of finite type. Additionally, f : X → X has shadowing if (f, X) lifts via a map which is ALP to the inverse limit of a sequence satisfying the Mittag- Leffler condition and consisting of shifts of finite type. The approach we take is topological rather than metric as this seems to provide the most natural proofs and allows for simple generalizations. Although we are considering inverse limits of dynamical systems, our techniques are very similar in flavour to the inverse limit of coupled graph covers which have been used by a number of authors to study dynamics on Cantor sets, for example [3, 11, 12, 13, 28, 29, 30]. The paper is arranged as follows. In Section 2, we formally define shadowing, shift of finite type and the inverse limit of a direct set of dynamical systems. In Section 3, we characterize shadowing as a topological, rather than metric property, and prove that an inverse limit satisfying the Mittag-Leffler condition which consists of systems with shadowing itself has shadowing (Theorem 8). Here we also introduce the orbit and pseudo-orbit shift spaces associated with a finite open cover of a dynamical system and observe in Theorem 12 that these capture the dynamics of f. Section 4 discusses compact, totally disconnected Hausdorff, but not necessarily metric, dynamical systems, showing that such systems have shadowing if and only if they are (conjugate to) the inverse limit of a directed system satisfying the Mittag- Leffler condition and consisting of shifts of finite type (Theorem 18). In Section 5, we examine the case of systems on general metric spaces, establishing in Theorem 20 a partial analogue to Theorem 18 and Corollary 19. In Section 6, we discuss factor maps which preserve shadowing and, in light of this, we are able to partially characterize compact metric and Hausdorff systems with shadowing in Theorem 26 and Corollary 28.

2. Preliminaries and Definitions By map, we mean a continuous function. The set of natural numbers, including 0, is denoted by ω. A dynamical system, (f, X), consists of a topological space X and a map f : X → X. In what follows, X need not necessarily be metric, but will typically be compact Hausdorﬀ. Given two dynamical systems (f, X) and (g, Y ), a factor map (or semiconjugacy) from (f, X) to (g, Y ) is a map φ : X → Y that commutes with the dynamics, i.e. φ◦f = g ◦φ. In this case, we say that the system (g, Y ) lifts via φ to the system (f, X). A factor map which is a homeomorphism is called a conjugacy. Deﬁnition 1. Let X be a compact metric space and let f : X → X be a continuous function. Let hxiii∈ω be a sequence in X. Then hxiii∈ω is a δ-pseudo-orbit provided d(xi+1, f(xi)) < δ for all i ∈ ω and the point z -shadows hxiii∈ω provided i d(xi, f (z)) < for all i ∈ ω. The map f has shadowing (or the pseudo-orbit tracing property) provided that for all > 0 there exists δ > 0 such that every δ-pseudo-orbit is -shadowed by a point. 4 C. GOOD AND J. MEDDAUGH

A particularly nice characterization of shadowing exists if we restrict our attention to shift spaces. For a ﬁnite set Σ, the full one-sided shift with alphabet Σ consists of the space of inﬁnite sequence in Σ, i.e. Σω using the product topology on the discrete space Σ and the shift map σ, given by

σhxii = hxi+1i. A shift space is a compact invariant subset X of some full-shift. A shift space X is a shift of finite type over alphabet Σ if there is a finite collection F of finite ω words in Σ for which hxii ∈ Σ belongs to X if and only if for all i ≤ j, the word xixi+1 ··· xj ∈/ F. A shift of finite type is said to be N-step provided that the length of the longest word in its associated set of forbidden words F is N + 1. As mentioned above, a shift space has shadowing if and only if it a shift of finite type [35]. Inverse limit constructions arise in a variety of settings. Many of the results here hold for arbitrary (non-metric) compact Hausdorff spaces and so we consider inverse limits of dynamical systems taken along an arbitrary directed set. (Recall that (Λ, ≤) is a directed set provided ≤ is a transitive order for which any pair x, y has an upper bound x, y ≤ z.) The reader, however, will not miss much by assuming that the space is compact metric in which case the inverse limit may be indexed by N.

Definition 2. Let (Λ, ≤) be a directed set. For each λ ∈ Λ, let Xλ be a compact η Hausdorff space and, for each pair λ ≤ η, let gλ : Xη → Xλ be a continuous map. η Then (gλ,Xλ) is called an inverse system provided that λ (1) gλ is the identity map, and ν η ν (2) for λ ≤ η ≤ ν, gλ = gλ ◦ gη . η The inverse limit of (gλ,Xλ) is the space η η lim{g ,Xλ} = {hxλi ∈ ΠXλ : ∀λ ≤ η xλ = g (xη)} ←− λ λ with topology inherited as a subspace of the Tychonoff product ΠXλ. Since the inverse limit of compact Hausdorff spaces is a closed subset of the product space, it is itself compact and Hausdorff. The following easily proved fact η is often useful. If U ⊆ lim{g ,Xλ} is open, and x ∈ U, then there exists λ and ←− λ −1 η Uλ ⊆ Xλ open with x ∈ π (Uλ) ∩ lim{g ,Xλ} ⊆ U. That is, the collection of sets λ ←− λ −1 η η of the form π (Uλ)∩lim{g ,Xλ} for Uλ open in Xλ forms a basis for lim{g ,Xλ}. λ ←− λ ←− λ η Additionally, it is also worth noting that, in this formulation, if the bonding maps gλ η η are surjective, then the restricted projection maps πγ | lim{g ,Xλ} : lim{g ,Xλ} → ←− λ ←− λ Xγ are also surjective. This is easily observed as follows. Fix γ ∈ Λ and z ∈ Xγ . For µ µ ≥ γ, define Aµ = {hxλi ∈ ΠXλ : ∀λ ≤ µ xλ = gλ (xµ)} ∩ {hxλi ∈ ΠXλ : xγ = z}. Since each gη is surjective, this is nonempty and compact. Furthermore, if ν ≥ µ, λ T it is clear that Aν ⊆ Aµ. Hence, the intersection A = Aµ is nonempty, consists only of points belonging to the inverse limit, and has πγ (A) = z. Now, suppose that for each λ in the directed set Λ, fλ : Xλ → Xλ is a continuous η function. If the bonding maps gλ commute with the functions fλ, then we can extend this definition to the family of dynamical systems {(fλ,Xλ): λ ∈ Λ}. Specifically we make the following definition. SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 5

Definition 3. Let (Λ, ≤) be a directed set. For each λ ∈ Λ, let (fλ,Xλ) be a dynamical system on a compact Hausdorff space and, for each pair λ ≤ η, let η η gλ : Xη → Xλ be a continuous map. Then (gλ, (fλ,Xλ)) is called an inverse system provided that λ (1) gλ is the identity map, and ν η ν (2) for λ ≤ η ≤ ν, gλ = gλ ◦ gη , and η η (3) for λ ≤ η, fλ ◦ gλ = gλ ◦ fη. η η The inverse limit of (g , (fλ,Xλ)) is the dynamical system ((fλ)∗, lim{g ,Xλ}), λ ←− λ where (fλ)∗ is the induced map given by (fλ)∗ hxλi = fλ(xλ) . Q Note that (fλ)∗ is the restriction of the product map fλ to the inverse limit η lim{g ,Xλ} and is, therefore, continuous. Moreover, it is easy to check that (fλ)∗ ←− λ η maps the inverse limit into itself, and thus ((fλ)∗, lim{g ,Xλ}) is indeed a contin- ←− λ uous dynamical system. Given a map f : X → X from a compact metric or Hausdorff space to itself, one is frequently interested in the inverse limit space lim(X, f) = {(x ): f(x ) = x } ←− i i+1 i under the action of the shift map σhxii = hxi+1i. We note that such spaces are a n+1 special case of Definition 3 applied with Λ = ω, Xn = X and fn = gn = f for all n ∈ ω. An argument similar to that for the surjectivity of the restricted projection η maps demonstrates that if each of the bonding maps gλ and each of the maps fλ is surjective, then the induced map (fλ)∗ is also surjective. However, although many of the inverse limits under consideration in this paper do not have surjective bonding maps, they do satisfy a less stringent condition. Definition 4. An inverse system (of spaces or of dynamical systems) satisfies the Mittag-Leffler condition provided that for all λ ∈ Λ, there exists γ ≥ λ such that γ η for each η ≥ γ, we have gλ(Xγ ) = gλ(Xη). For simplicity of notation going forward, we will say that an inverse system is an ML inverse system if it satisfies the Mittag-Leffler condition. The Mittag-Leffler condition for inverse sequences was defined by Grothendieck [15, Definition 13.1.2] although it is implicit in Bourbaki [4, Chapter II, Theorem 1] (see, for example, [16] for more on the ML condition). We note that an inverse system with surjective bonding maps automatically satisfies the Mittag-Leffler condition. Moreover, in a system satisfying the Mittag- γ Leffler condition, if γ witnesses the condition with respect to µ and x ∈ gµ(Xγ ) ⊆ −1 η Xµ, then πµ (x) ∩ lim{g ,Xλ}= 6 . ←− λ ∅ It is well known that for any inverse system there is an inverse system satisfying the Mittag-Leffler condition (in fact, one with surjective bonding maps) which has the same inverse limit. To see this, for each factor space Xλ define ˜ T η η ˜ ˜ Xλ = η>λ gλ(Xη) and define the bonding mapg ˜λ : Xη → Xλ to be the restric- η ˜ tion of gλ. The inclusion maps from Xλ → Xλ induce a map between the inverse limits which is easily seen to be a homeomorphism. However, in making this mod- ification, we lose information about the factor spaces. Indeed, every system on a compact, totally disconnected Hausdorff space is conjugate to an inverse limit of shifts of finite type. Thus, by the above argument, every system on a compact, 6 C. GOOD AND J. MEDDAUGH totally disconnected Hausdorff space is conjugate to the inverse limit of an inverse system satisfying the Mittag-Leffler condition, but consisting of subshifts which may or may not be of finite type. As we shall see, not every such system has the shadowing property—only those systems which are conjugate to an inverse limit of an inverse system satisfying the Mittag-Leffler condition and consisting of shifts of finite type have the shadowing property.

3. Shadowing without Metrics Shadowing is on first inspection a metric property, and indeed the properties of metrics often play a role in its investigation and application. However, shadowing can be viewed as a strictly topological property, defined in terms of finite open covers, provided that we restrict our attention to compact metric spaces. Similar observations have been made in [8, 14]. We assume in what follows that the elements of a cover U are non empty open sets. Definition 5. Let X be a space, let f : X → X, and let U be a finite open cover of X.

(1) The sequence hxiii∈ω is a U-pseudo-orbit provided for every i ∈ ω, there exists Ui+1 ∈ U with xi+1, f(xi) ∈ Ui+1. (2) Let hUii be a sequence of elements of U. We say that hUii is a U-pseudo- orbit pattern provided there is a sequence hxii of points in X such that xi+1, f(xi) ∈ Ui+1 for each i. We say that hUii is a U-orbit pattern provided i there is some z such that f (z) ∈ Ui for all i. (3) The point z ∈ X U-shadows hxiii∈ω provided for each i ∈ ω there exists i Ui ∈ U with xi, f (z) ∈ Ui. Lemma 6. Let X be a compact metric space. Then f : X → X has shadowing if and only if for every finite open cover U, there exists a finite open cover V, such that every V-pseudo-orbit is U-shadowed by some point z ∈ X. Proof. First, suppose that f has the shadowing property and let U be a finite open cover of X. Fix > 0 so that for each -ball B in X, there exists U ∈ U with B ⊆ U. Now, let δ > 0 witness -shadowing. Let V be a finite open cover of X refining U which consists of open sets of diameter less that δ. Now, Let hxii be a sequence in X as in the statement of the lemma. Then d(xi, f(xi−1)) < diam(Vi) < δ for all i ∈ ω \ 0. In particular, hxii is a δ-pseudo- i orbit. Let z ∈ X be an -shadowing point for this sequence. Then d(xi, f (z)) < i for each i ∈ ω, and in particular, {xi, f (x)} ⊆ B(xi). By construction, there i exists Ui ∈ U for which {xi, f (x)} ⊆ B(xi) ⊆ Ui, satisfying the conclusion of the lemma. Conversely, let us suppose that f satisfies the open cover condition of the lemma. Let > 0, and consider a finite subcover U of X consisting of /2-balls. Let V be the cover that witnesses the satisfaction of the condition, and choose δ > 0 such that for each δ-ball in X, there is an element of V which contains it. Now, fix a δ-pseudo-orbit hxii. Then for each i ∈ ω \ 0, d(xi, f(xi−1)) < δ, and hence there exists Vi ∈ V such that xi, f(xi−1) ∈ Vi. Let z ∈ X be the point guaranteed by the open cover condition. Then, for each i ∈ ω, there exists i i Ui ∈ U with xi, f (z) ∈ Ui. But Ui is an /2-ball and hence d(xi, f (x)) < , i.e. z -shadows the pseudo-orbit. SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 7

This observation allows the decoupling of shadowing from the metric, and we can then take the following definition of shadowing, which is valid for systems with compact Hausdorff (but not necessarily metric) domain, an application that has recently seen increased interest [7, 8, 14]. Definition 7. Let X be a (nonempty) compact Hausdorff topological space. The map f : X → X has shadowing provided that for every finite open cover U, there exists a finite open cover V such that every V-pseudo-orbit is U-shadowed by a point of X. Clearly if the cover V witnesses U-shadowing, the cover V0 refines V and the cover U 0 is refined by U, then V0 witnesses U 0-shadowing. With this definition in mind, we can prove the following result which will be important to the characterization of shadowing in Section 4. Theorem 8. Let f : X → X be conjugate to an ML inverse system consisting of maps with shadowing on compact spaces. Then f has shadowing. λ Proof. Without loss, let (Λ, ≤) be a directed set and (f, X) = lim{gγ , (fλ,Xλ)} ←− be an ML inverse system where each of (fλ,Xλ) is a system with shadowing on a compact space. λ Let U be a finite open cover of X. Since X = lim{gγ ,Xλ}, we can find λ and ←− −1 a finite open cover Wλ of Xλ so that W = {πλ (W ) ∩ X : W ∈ Wλ} refines U. γ η There is some γ ≥ λ such that for each η ≥ γ, we have gλ(Xγ ) = gλ(Xη). γ −1 Let Wγ = {(gλ) (W ): W ∈ Wλ}. Since fγ has shadowing, there is a finite non-empty open cover Vγ of Xγ such that every Vγ -pseudo orbit is Wγ -shadowed. −1 Let V = {πγ (V ) ∩ X : V ∈ Vγ } and let hxii be a V-pseudo-orbit with pat- −1 −1 tern hπγ (Vi) ∩ Xi. Then for each i ∈ ω, we have f(xi) ∈ f(πγ (Vi) ∩ X) ∩ −1 πγ (Vi+1) ∩ X 6= ∅. It follows then, that h(xi)γ i is a Vγ -pseudo-orbit with pattern hVii. Since every Vγ -pseudo orbit is Wγ -shadowed, there is some zγ ∈ Xγ and a i sequence hWii of elements from Wγ such that fγ (zγ ), (xi)γ ∈ Wi. Note that this T −i means that zγ ∈ fγ (Wi) 6= ∅. Now each Wi is the inverse image of some element of Wλ, so equivalently, we have 0 0 γ γ T −i a sequence hWi i ∈ Wλ with (xi)λ ∈ Wi and with zλ = gλ(zγ ) ∈ gλ( fγ (Wi)) ⊆ T −i 0 fλ (Wi ) 6= ∅. In particular, since this is an ML inverse system, there is some γ T −i T −i 0 z ∈ X with zλ ∈ gλ( fγ (Wi)) ⊆ fλ (Wi ). i −1 0 It then follows that f (z), xi ∈ πλ (Wi ) ∩ X, so that hxii is W-shadowed, and hence U-shadowed as required. We wish to capture the dynamics of the map f on X via action of the shift map induced by f on the space of orbit or pseudo-orbit patterns for certain covers of X. If U is an open cover then one can prove (as in Lemma 11) that the collection of all U-pseudo orbit patterns forms a closed subspace of the product space U ω of all sequences of elements from U (where U is given the discrete topology). This implies that the collection of all pseudo-orbit patterns of a finite open cover is a subshift of U ω (and is indeed a shift of finite type). However, the space of all orbit patterns need not even be closed subset of U ω. To see this consider the period doubling map θ 7→ 2θ mod 2π on the unit circle [0, 2π). Let U1 = (0, π) and U2 = (π − , 0 + ) n+1 for some suitably small . For each n, choose zn near to 0 such that f (zn) = π n+2 and f (z) = 0. Then zn generates an orbit pattern hVn,ii such that Vn,i is U1 8 C. GOOD AND J. MEDDAUGH for i ≤ n and U2 for i > n. Clearly the sequence of sequences hVn,ii converges to the constant sequence hU1,U1,U1,... i, which is not an orbit pattern for the cover ω {U1,U2}, so that the collection of orbit patterns is not closed in U . With this in mind, we have the following definitions. Definition 9. Let f : X → X be a map on the space X, let U be a finite open cover of X, each element of which is nonempty and let U ω be the one-sided shift space on the alphabet U with shift map σ. (1) The U-orbit space is the set O(U) ⊆ U ω which is the closure of the set consisting of all sequences hUii in U for which there exists z ∈ X with i f (z) ∈ Ui. (2) The U-pseudo-orbit space is the set PO(U) ⊆ U ω consisting of all sequences hUii in U for which there exists a sequence hxii with xi+1, f(xi) ∈ Ui+1. ω Additionally, for U ∈ U and i ∈ ω, define πi : U → U to be projection onto the i-th coordinate. The following lemma is immediate and provides an alternate description of O(U) and PO(U). Lemma 10. Let f : X → X be a map on X and let U be a finite open cover. Then

ω \ −i O(U) = {hUii ∈ U : f (Ui) 6= ∅} \ ω \ −i = {hUii ∈ U : f (Ui) 6= ∅} n∈ω i≤n and ω PO(U) = {hUii ∈ U : f(Ui) ∩ Ui+1 6= ∅}. As consequence, we have the following relations between O(U), PO(U) and U ω. Lemma 11. Let f : X → X be a map on X and let U be a finite open cover. Then, O(U) is a subset of PO(U) and both spaces are subshifts of U ω. In particular, PO(U) is a 1-step shift of finite type. Proof. That O(U) ⊆ PO(C) ⊆ U ω is immediate. It is also clear that each of these spaces is shift invariant. Hence, since O(U) is closed by definition, it is a subshift. That PO(U) is a 1-step subshift of finite type follows by observing that if hUii is not a pseudo-orbit pattern, then for some i, there f(Ui) ∩ Ui+1 = ∅, and so we can forbid hUii from PO(U) by forbidding the word UiUi+1. Clearly there are only finitely many such words. If X is compact Hausdorff, then the entire dynamics of a map f are encoded in the orbit spaces of an appropriate system of covers of X. In particular, let FOC(X) be the collection of all finite open covers of X. This collection is naturally partially ordered by refinement and forms a directed set. Theorem 12. Let f : X → X be a map on the compact Hausdorff space X. Let {Uλ}λ∈Λ be a cofinal directed subset of FOC(X). Then for all x ∈ X there exists a T choice of Uλ(x) ∈ Uλ with {x} = Uλ(x) and furthermore for any such sequence, we have for all n ∈ ω, n \ n −1 {f (x)} = π0 σ O(Uλ) ∩ π0 (Uλ(x)) . SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 9

Proof. Let f, Λ, and {Uλ} be as described. Fix x ∈ X. For each λ ∈ Λ, choose T Uλ(x) ∈ Uλ with x ∈ Uλ(x). Then x ∈ Uλ(x). Furthermore, for all y ∈ X \{x}, there exists a cover U of X such that if x ∈ U ∈ U, then y∈ / U. Then for any λ ∈ Λ T with Uλ reﬁning U, y∈ / Uλ(x) regardless of choice of Uλ(x), hence y∈ / Uλ(x) and T therefore y∈ / Uλ. n −1 Now, for each λ, it is straightforward to show that π0 σ O(Uλ) ∩ π0 (Uλ(x)) n n T n −1 is equal to f (Uλ(x)). In particular, f (x) ∈ π0 σ O(Uλ) ∩ π0 (Uλ(x)) . T n −1 Suppose now that z ∈ π0 σ O(Uλ) ∩ π0 (Uλ(x)) . Then for each λ, there n exists xλ ∈ Uλ(x) with z = f (xλ). But, by construction, x is a limit point of {xλ}, n n T n −1 and by continuity, z = f (x). Hence {f (x)} = π0 σ O(Uλ) ∩ π0 (Uλ(x)) as claimed.

It should be noted that for general Hausdorﬀ spaces, the structure of {Uλ} may be quite complex, but for metric X, it is the case that a sequence of covers will always suﬃce and we will make use of this fact in the following sections. In the metric case, Theorem 12 is equivalent to Theorem 3.9 of [28], although that result is expressed in terms of graph covers and relations.

4. Characterizing shadowing in totally disconnected spaces In the sense of Theorem 12, the entire dynamics are encoded by the action of f on an appropriate collection of refining covers. This is not unlike the way that the topology is completely encoded as well. In this section we explore this analogy. In particular, it is well known that a space X is chainable, i.e. can be encoded with a sequence of refining chains (i.e. finite covers with Ui ∩ Uj 6= ∅ if and only if |i − j| ≤ 1) if and only if X can be written as an inverse limit of arcs. In a sense, the arc is the fundamental chainable object. In an analogous fashion we show that shifts of finite type are the fundamental objects among dynamical systems on totally disconnected spaces with shadowing. Without loss of generality, in the case that X is totally disconnected compact Hausdorff, the cofinal directed subset of FOC(X) in Theorem 12 can be taken to consist of open covers which are each finite collections of pairwise disjoint open (and hence also closed) sets. For the purposes of the following, we refer to such finite pairwise disjoint open covers as partitions of X. Let U and V be arbitrary covers of X with V refining U. Then let ι : V → U be defined so that V ∩ ι(V ) 6= ∅. In the case that U and V are both partitions, this is equivalent to asking V ⊆ ι(V ), so that ι is a well-defined function. In general, if U and V are not partitions, ι is a multifunction and it is this that creates the obstacle to dealing with non-totally disconnected spaces. We address this issue in Section 5. When considering partitions, the map ι naturally induces a continuous map ι : Vω → U ω, the domain of which can then be restricted to O(V) or PO(V) as appropriate. As the intended domain is typically clear, the symbol ι will be used for all. Note that if U is a partition of X, its elements are necessarily compact, and in this case, it is easy to see that the set of orbit sequences is naturally closed, so that

ω \ −i O(U) = {hUii ∈ U : f (Ui) 6= ∅}. 10 C. GOOD AND J. MEDDAUGH

Lemma 13. Let f : X → X be a map on the compact, totally disconnected Haus- dorff space X and let U and V be partitions of X with V refining U. Then σ and ι commute and the following statements hold: (1) ι(O(V)) = O(U). (2) O(U) ⊆ ι(PO(V)) ⊆ PO(U). Proof. It is immediate from their definitions that σ and ι commute on their unre- stricted domains. Towards proving statement (1), consider hVii ∈ O(V). Then there is x such that i i f (x) ∈ Vi for all i. But then f (x) ∈ ι(Vi), so that hι(Vi)i ∈ O(U). Conversely, T −i if hUii ∈ O(U), then we can choose x ∈ f (Ui) 6= ∅. Now choose hVii so that −1 x ∈ f (Vi) for each i. Clearly hVii ∈ O(V). Since V refines U and the elements of U are pairwise disjoint and clopen, it follows that Vi ⊆ Ui, i.e. hUii = ιhVii. Statement (2) follows similarly. The additional structure of totally disconnected spaces allows us to state the following immediate corollary to Theorem 12. In particular, the collection Part(X) of partitions of X is a cofinal directed subset of FOC(X). Since the elements of the covers in Part(X) are closed, the refinement relations of Part(X) are in fact closure refinements, so all nested intersections are nonempty. Corollary 14. Let f : X → X be a map on the compact Hausdorff totally disconnected space X. Let {Uλ}λ∈Λ be a cofinal directed suborder of Part(X). Then the system (f, X) is conjugate to (σ∗, lim{ι, O(Uλ)}) by the map ←− \ hwλi 7→ π0(wλ). It is important to note that the maps ι in the inverse system depend very much on their domain and range. However, if W refines V which in turn refines U, then the composition of ι : W → V and ι : V → U is precisely the same as ι : W → U, and as such the inverse system is indeed well-defined. The existence of partitions also allows us to state the following alternative characterization of shadowing. Lemma 15. Let f : X → X be a map on the compact Hausdorff totally disconnected space X. Then f has shadowing if and only if for each U ∈ Part(X), there exists V ∈ Part(X) which refines U such that for all W ∈ Part(X) which refine V, ι(PO(W)) = O(U). Proof. Let f have shadowing and let U ∈ Part(X). Let V be the cover witnessing shadowing. Without loss of generality, V ∈ Part(X) and V refines U. Now, let hxii be a V-pseudo-orbit with hVii its V-pseudo-orbit pattern, and let z ∈ X be i a shadowing point with hUii its shadowing pattern. By definition, xi and f (z) belong to Ui, and hence Vi ∩ Ui 6= ∅, and since V refines U and the elements of U are disjoint, we have Vi ⊆ Ui, i.e. ι(Vi) = Ui and hence ιhVii = hUii. Thus ι(PO(V)) ⊆ O(U), and the reverse inclusion is given by Lemma 13, and thus the two sets are equal. Now, for any W ∈ Part(X) which refines V, observe that O(U) ⊆ ι(PO(W)) ⊆ ι(PO(V)) = O(U), Conversely, suppose that f has the stated property regarding open covers. Let U be a finite open cover of X. Since X is totally disconnected, let U 0 ∈ Part(X) SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 11 which refines U. Let V be the cover witnessing the property with respect to U 0. Now, let hxii be a V-pseudo-orbit and let hVii be its V-pseudo-orbit pattern. By 0 0 0 T −i 0 the property, there exists hUi i ∈ O(U ) with ιhVii = hUi i. Now, let z ∈ f (Ui ). i 0 Then xi, f (z) ∈ Ui which in turn is a subset of some Ui ∈ U. In particular, z U-shadows the V-pseudo-orbit hxii. In light of this, we have the following theorem. Theorem 16. Let f : X → X be a map with shadowing on the compact totally disconnected Hausdorff space X. Let {Uλ}λ∈Λ be a cofinal directed subset of Part(X). Then the system (σ∗, lim{ι, O(Uλ)}) is conjugate to (σ∗, lim{ι, PO(Uλ)}) and ←− ←− both systems satisfy the Mittag-Leffler condition.

Proof. First, observe that for each λ, O(Uλ) is a subset of PO(Uλ). It is a standard result in inverse limit theory that the map j∗ : lim{ι, O(Uλ)} → lim{ι, PO(Uλ)}} ←− ←− induced by inclusion is a continuous injection, and clearly commutes with σ∗. In fact, this is a surjection, and hence demonstrates the desired conjugacy. This is easily proven by considering the following. Define a monotone function p :Λ → Λ such that for each λ ∈ Λ, we have p(λ) ≥ λ and ι(PO(Up(λ))) = O(Uλ). Then, define the map φ : lim{ι, PO(Uλ)} → ←− lim{ι, O(Uλ)} as follows. ←− φ(hwγ i)λ = ι(wp(λ)) That this is well-defined and continuous is a standard result in inverse limit theory. As this is induced by the maps ι, it will commute with σ∗. Now, consider j∗ ◦ φ : lim{ι, PO(Uλ)} → lim{ι, PO(Uλ)}. Consider hwγ i ∈ ←− ←− lim{ι, PO(Uλ)}. We see that ←−

(j∗ ◦ φ(hwγ i))λ = jλ(ι(wp(λ))) = jλ(wλ) = wλ

In particular, j∗ ◦ φ is the identity on lim{ι, PO(Uλ)}, and since j∗ is injective, ←− it follows that both j∗ and φ are conjugacies. It remains to be shown that these systems satisfy the Mittag-Leffler condition. For the system O(Uλ), note that the inclusion maps are surjective by Lemma 13, and hence the system has the condition. For the system PO(Uλ), we proceed as follows. Let λ ∈ Λ and choose γ ≥ λ so that Uγ witnesses Uλ shadowing. Then for η γ all η ≥ γ, by Lemma 15, we have ιλ(PO(Uη)) = O(Uλ) = ιλ(PO(Uγ )). This theorem complements Corollary 14, and by applying Lemma 11, and the well-known fact that shifts of finite type have shadowing [35], we have the following result. Corollary 17. Let f : X → X be a map with shadowing on the compact totally disconnected Hausdorff space X. Then (f, X) is conjugate to an inverse limit of an ML inverse system of shifts of finite type. In fact, this is a complete characterization of totally disconnected systems with shadowing; the following is an immediate consequence of Corollary 17 and Theorem 8. 12 C. GOOD AND J. MEDDAUGH

Theorem 18. Let X be a compact, totally disconnected Hausdorff space. The map f : X → X has shadowing if and only if (f, X) is conjugate to the inverse limit of an ML inverse system of shifts of finite typ. Of course, Theorem 18 includes metric systems. However, if X is metric, we may easily find sequences {Un}n∈N of partitions which are cofinal directed suborders of Part(X). In particular, we can let U0 = {X}, and for each Ui, let Ui+1 be a partition −1 of X with mesh less than 2 which refines Ui and which witnesses shadowing for Ui. Then the function p from the proof of Theorem 16 simply increments its input. The conjugacy then follows from the induced diagonal map ι∗ on the inverse systems as seen in Figure 1.

ι ι ι ι lim{ι, PO(Ui)} ←− PO(U0) PO(U1) PO(U2) PO(U3) ··· ι ι ι ι ι∗ ' j j j j

ι ι ι ι lim{ι, O(Ui)} ←− O(U0) O(U1) O(U2) O(U3) ···

Figure 1. Diagram for the metric case of Theorem 18

This observation immediately implies the following. Corollary 19. Let X be the Cantor set, or indeed any compact, totally disconnected metric space. The map f : X → X has shadowing if and only if (f, X) is conjugate to the inverse limit of an ML sequence of shifts of ﬁnite type. This ad hoc construction of an appropriate sequence of covers can be modiﬁed into a technique that will apply to general compact metric spaces in Section 5.

5. Shadowing in General Metric Systems Theorem 12 applies equally well to systems in which there are non-trivial connected components, and as such, one might hope for analogue to Corollary 14. However, as mentioned, the principal obstruction to a direct application of the methods of Section 4 is that the intersection relation ι is no longer necessarily single- valued, so that the induced map on the pseudo-orbit space is not only set-valued, but also not finitely determined. However, by modifiying the approach illustrated in Figure 1, we are obtain the following. Recall that for a cover C of X and A ⊆ X, the star of A in C is the set st(A, C) which is the union of all elements of C which meet A. Theorem 20. Let X be a compact metric space and f : X → X be a continuous n+1 map with shadowing. Then there is an inverse sequence (gn ,Xn) of shifts of finite n+1 type such that (f, X) is a factor of (σ∗, lim{gn ,Xn}). ←− Proof. First, observe that since X is compact metric and f has shadowing, we can easily find a sequence hUii of finite open covers satisfying the following properties:

(1) Un+1 witnesses Un shadowing, (2) {Ui} is coﬁnal in FOC(X), and (3) for all U ∈ Un+2, there exists W ∈ Un such that st(U, Un+1) ⊆ W . SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 13

This is easily accomplished by taking U0 = {X}, and inductively letting Un+1 be a cover witnessing Un-shadowing with mesh less than one third the Lebesgue number of the cover Un. Conditions (1) and (2) are immediately met. To verify that condition (3) is satisfied, fix n ∈ N and U ∈ Un+2. Then U is a subset of V for some V ∈ Un+1, and so st(U, Un+1) is a subset of st(V, Un+1). But the diameter of st(V, Un+1) is at most three times the mesh of Un+1, and hence has diameter less than the Lebesgue number for Un. Hence there is some W ∈ Un for which W ⊇ st(V, Un+1) ⊇ st(V, Un+1) as required. Let f : X → X and covers hUii be as stated. For each U ∈ Un+2, fix W (U) ∈ Q Un with st(U, Un+1) ⊆ W (U), and define w : PO(Un+2) → Un by w(hUji) = hW (Uj)i. Note that, as this is a single letter substitution map on a shift space, it is a continuous map and commutes with the shift map by definition. We claim that w(PO(Un+2)) is a subset of O(Un). Indeed, let hUji ∈ PO(Un+2) and hxji a pseudo-orbit with this pattern. Since Un+2 witnesses Un+1-shadowing, j there exists z ∈ X and sequence hVji ∈ O(Un+1) with f (z), xj ∈ Vj. In particular, for any such z and choices of hVji, Vj ⊆ st(Uj, Un+1) ⊆ W (Uj). Indeed, this establishes that hW (Uj)i ∈ O(Un). It should be noted that while w is not necessarily j surjective, for every x ∈ X, there is some hUji in w(PO(Un+2)) with f (x) ∈ Uj j for all j. We can observe this by noting that hf (x)i is itself a Un+2-pseudo-orbit, j and in particular, we have f (x) ∈ W (Uj), and so hW (Uj)i is a Un orbit pattern for x. Since O(Un) ⊆ PO(Un), we have the following diagram (Figure 2).

PO(U0) PO(U1) PO(U2) PO(U3) PO(U4)

w w

O(U0) O(U1) O(U2) O(U3) O(U4)

Figure 2. Diagram for the proof of Theorem 20

So, while the ‘natural’ map from PO(Un+2) is set-valued, the composition of inclusion and w gives a single-valued continuous map from O(Un+2) to O(Un), and by reversing the order of composition, from PO(Un+2) to PO(Un). We will denote 0 these maps by ι . Figure 3 then establishes the existence of a map w∗ from the inverse limit of the pseudo-orbit spaces ti the inverse limit of the orbit spaces which commutes with the induced maps σ∗. All that remains is to establish that the inverse limit of orbit space is a factor of the system (f, X) and that the composition of this factor map with the map 0 w∗ is a surjection. For the former, Let φ : lim{ι , O(U2i)} → X be given by ←− T φhuii = π0(ui). Note that, by construction, πk(ui+1) ⊆ πk(ui) for all k and i, so in particular φ(huii) is a nested intersection of the closures of elements of the open covers, and hence is well-deﬁned. That φ is continuous and commutes with σ∗ follows from similar reasoning as Theorem 12 and Corollary 14. That φ is surjective follows from the same logic as Theorem 12–observe that that for all x ∈ X, and all k, there exists a nonempty subset O2k(x) of U2k-orbit patterns for x in O(U2k), 14 C. GOOD AND J. MEDDAUGH

0 0 0 0 0 ι ι ι ι lim{ι , PO(Ui)} ←− PO(U0) PO(U2) PO(U4) PO(U6) ··· w w w w w∗ '

0 0 0 0 0 ι ι ι ι lim{ι , O(U2i)} ←− O(U0) O(U2) O(U4) O(U6) ···

Figure 3. Diagram for the proof of Theorem 20 and that \ −1 \ 0 O(x) = π (O2k) lim{ι , O(U2i)} 2k ←− is nonempty and satisfies φ(O(x)) = {x}. This same observation coupled with the fact that O(U2k) is mapped into PO(U2k) by inclusion demonstrates that φ ◦ w∗ is indeed surjective, and is thus a factor map. Clearly, the existence of the sequence of covers satisfying conditions (2) and (3) in this proof is a strong condition. In particular, this implies that there is such a cofinal sequence in FOC(X), which in turn implies that the space X is metrizable. While it seems that this might be straightforward to generalize, a direct general- ization of this argument fails due to the complexity of the partial order on the class of open covers of a general compact Hausdorff space. In particular, given any pair U and V of finite open covers of a compact Hausdorff space X, there is some cover W which mutually star-refines U and V and also a cover T which is star-refined by both. To follow the argument from before, we would need to be able to establish maps from W to U and to V as well as maps from U and V into T all of which respect star-refinement and such that the compositions agree. This is generally not possible due to the inherent ‘drift’ of stars of sets in covers.

6. Factor maps which preserve shadowing We have now established that for a metric system to exhibit shadowing, it is necessary for there to exist an inverse limit of an inverse sequence of shifts of finite type of which the original system is a factor. However, it is worth noting that this is by no means sufficient, even with the added hypothesis that the inverse sequence is an ML inverse sequence. In particular, every sofic shift is a factor of such an inverse limit, but only those that are shifts of finite type exhibit shadowing.

Example 21. Let X be the subshift of {0, 1}Z consisting of those bi-infinite words containing at most one 1. The system (σ, X) fails to have shadowing, but is a factor of the inverse limit of an inverse sequence of shifts of finite type. Proof. (σ, X) is a standard example of a system which does not have shadowing; it has only one non-constant full orbit, namely the orbit passing through ··· 0001000 ··· , but uncountably many distinct pseudo-orbits containing the fixed point h0i. Now let Y be the subshift of {0, 1, 2}Z consisting of those sequences in which the words 02, 10, 21 and 20 do not appear, i.e. Y is the subshift of all sequences of the form ··· 0000001222222 ··· along with the constant sequences h0i SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 15 and h2i. Y is a shift of finite type. Then Y is (trivially) an inverse limit of an ML system of shifts of finite type. However, the map from Y to X induced by substituting the symbol 0 for 2 witnesses that X is a factor of Y . It is then natural to ask which factors of inverse limits of ML inverse systems consisting of shifts of finite type exhibit shadowing, i.e. is there a class of maps P such that if (f, X) is a factor by a map in P of an inverse limit of an ML inverse system of shifts of finite type, then (f, X) has shadowing? Of course, it is clear that there is such a class, and in fact the class of homeomorphisms have this property, but we wish to find, if possible, the maximal such class. Towards this end, we define the following. Definition 22. Let (f, X) and (g, Y ) be dynamical systems with X and Y compact Hausdorff spaces. A factor map φ :(f, X) → (g, Y ) lifts pseudo-orbits provided that for every VX ∈ FOC(X), there exists VY ∈ FOC(Y ) such that if hyii is a VY - pseudo-orbit in Y , then there is a VX -pseudo-orbit hxii in X with hyii = hφ(xi)i. Theorem 23. Let (f, X) and (g, Y ) be dynamical systems with X and Y compact Hausdorff. If (f, X) has shadowing and φ :(f, X) → (g, Y ) is a factor map that lifts pseudo-orbits, then (g, Y ) has shadowing.

Proof. Fix an open cover UY ∈ FOC(Y ), and let UX ∈ FOC(X) such that φ(UX ) refines UY . Since (f, X) has shadowing, let VX ∈ FOC(X) witness shadowing with respect to UX . Since φ lifts pseudo-orbits, let VY witness this with respect to VX . Finally, let hyii be a VY -pseudo-orbit. Pick hxii to be a VX -pseudo-orbit with hφ(xi)i = hyii. As every VX -pseudo-orbit is UX -shadowed, fix zX ∈ X to witness this and let zY = φ(zX ). It then follows that i for each i, we have φ (zX ), φ(xi) ∈ φ(UX,i) for some Ui ∈ UX . As φ(UX ) refines i UY , it follows that there exists UY,i ∈ UY with φ (zY ), yi ∈ φ(UY,i), i.e. zY = φ(zX ) UY -shadows hyii. Notwithstanding Theorem 23, a more general concept of lifting pseudo-orbits provides a much sharper insight into the relation between shadowing and shifts of finite type in compact metric systems. Definition 24. Let (f, X) and (g, Y ) be dynamical systems with X and Y compact Hausdorff spaces. A factor map φ :(f, X) → (g, Y ) almost lifts pseudo-orbits (or f is an ALP map) provided that for every VX ∈ FOC(X) and WY ∈ FOC(Y ), there exists VY ∈ FOC(Y ) such that if hyii is a VY -pseudo-orbit in Y , then there is a VX -pseudo-orbit hxii in X such that for each i ∈ N there exists Wi ∈ WY with φ(xi), yi ∈ Wi. Theorem 25. Let (f, X) and (g, Y ) be dynamical systems with X and Y compact Hausdorff and let φ :(f, X) → (g, Y ) be a factor map. Then the following statements hold: (1) if (f, X) has shadowing and φ is an ALP map, then (g, Y ) has shadowing, and (2) if (g, Y ) has shadowing then φ is an ALP map. Proof. First, we prove statement (1). Let (f, X) have shadowing and let φ be an ALP map. Fix an open cover UY ∈ FOC(Y ). Let WY ∈ FOC(Y ) such that if 16 C. GOOD AND J. MEDDAUGH

0 0 0 W, W ∈ WY with W ∩ W 6= ∅, then there exists U ∈ UY with W ∪ W ⊆ U, and let UX ∈ FOC(X) such that φ(UX ) refines WY . Since (f, X) has shadowing, let VX ∈ FOC(X) witness shadowing with respect to UX . Since φ is ALP, let VY witness this with respect to WY and VX . Finally, let hyii be a VY -pseudo-orbit. Pick hxii to be a VX -pseudo-orbit so that hφ(xi)i WY -shadows hyii. As every VX -pseudo-orbit is UX -shadowed, fix zX ∈ X to witness this and let zY = φ(zX ). i It then follows that for each i, we have φ (zX ), φ(xi) ∈ φ(UX,i) for some Ui ∈ UX . i As φ(UX ) refines WY , it follows that there exists Wi ∈ WY with φ (zY ), φ(xi) ∈ 0 Wi. Additionally, as pseudo-orbits are almost lifted, there exists Wi ∈ WY with 0 0 φ(xi), yi ∈ Wi . In particular, φ(xi) ∈ Wi ∩ Wi , so we have that there exists i 0 UY,i ∈ UY with φ (zY ), φ(xi), yi ∈ Wi ∪ Wi ⊆ UY,i ∈ UY , i.e zY UY -shadows hyii. Now, to prove statement (2), assume that (g, Y ) has shadowing. let VX ∈ FOC(X) and WY ∈ FOC(Y ). Let VY ∈ FOC(Y ) witness shadowing with respect to WY . Now, let hyii be a VY pseudo-orbit in Y . Then there is some z ∈ Y −1 i with z WY -shadowing hyii. Now, choose x ∈ φ (z) and observe that hf (x)i is i i a VX -pseudo-orbit (as it is in fact an orbit), and φ(f (x)) = g (z), so there exists i Wi ∈ WY with φ(f (z)), yi ∈ Wi (as given by the shadowing pattern of z and hyii). Thus φ is an ALP map. One consequence of the above is that the conjugacies (factor maps) in Theorems 18 and 20 are ALP maps. This is not terribly surprising in the case of Theorem 18, as the map in question is a homeomorphism, but in the case of Theorem 20, this is interesting, and allows us to refine the characterization. It should be also be noted that, as a result of this theorem, we see that the factor maps constructed by Bowen in [6] are ALP. Theorem 26. Let X be a compact Hausdorff space. The map f : X → X has shadowing if (f, X) lifts via a map which is ALP to the inverse limit of an ML inverse system of shifts of finite type.

Proof. This follows immediately from Theorem 8 and Theorem 25. In the metric case, we can say a bit more, but first we first translate the notion of almost lifting pseudo-orbits from the language of covers into the language of metric spaces. This is not completely necessary, but allows for a different perspective on the property. As this is a direct translation and application of Theorem 20 and Theorem 26, we state the following results without proof. Lemma 27. Let (f, X) and (g, Y ) be dynamical systems with X and Y compact metric spaces. A factor map φ :(f, X) → (g, Y ) is an ALP map if and only if for all > 0 and η > 0, there exists δ > 0 such that if hyii is a δ-pseudo-orbit in Y , there exists an η-pseudo-orbit hxii in X with d(φ(xi), yi) < . Corollary 28. Let X be a compact metric space. The map f : X → X has shadowing only if (f, X) is a factor of an inverse limit of a sequence of shifts of finite type by a map which is ALP. Additionally, f : X → X has shadowing if (f, X) lifts via a map which is ALP to the inverse limit of an ML sequence of shifts of finite type. Of course, it would be of significant benefit if ALP maps had an alternate characterization. In particular, it is clear that homeomorphisms and covering maps lift SFTS AS FUNDAMENTAL OBJECTS IN THE THEORY OF SHADOWING 17 pseudo-orbits. However, there are maps which are neither covering maps nor home- morphisms which almost lift pseudo-orbits, in particular, the factor maps given in Theorem 20 are not typically open, much less covering maps.

7. Acknowledgement The authors are grateful to the referee for the detailed and insightful comments on the previous version of this paper.

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(C. Good) University of Birmingham, School of Mathematics, Birmingham B15 2TT, UK Email address, C. Good: [email protected]

(J. Meddaugh) Baylor University, Department of Mathematics, Waco TX, 76798 USA Email address, J. Meddaugh: jonathan [email protected]